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Theorem sepnsepolem2 47854
Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 47855. Proof could be shortened by 1 step using ssdisjdr 47792. (Contributed by Zhi Wang, 1-Sep-2024.)
Hypothesis
Ref Expression
sepnsepolem2.1 (πœ‘ β†’ 𝐽 ∈ Top)
Assertion
Ref Expression
sepnsepolem2 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
Distinct variable groups:   𝑦,𝐷   𝑦,𝐽   π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦)   𝐷(π‘₯)   𝐽(π‘₯)

Proof of Theorem sepnsepolem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sepnsepolem2.1 . 2 (πœ‘ β†’ 𝐽 ∈ Top)
2 id 22 . . 3 (𝐽 ∈ Top β†’ 𝐽 ∈ Top)
3 sslin 4230 . . . . 5 (𝑧 βŠ† 𝑦 β†’ (π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦))
4 sseq0 4395 . . . . . 6 (((π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦) ∧ (π‘₯ ∩ 𝑦) = βˆ…) β†’ (π‘₯ ∩ 𝑧) = βˆ…)
54ex 412 . . . . 5 ((π‘₯ ∩ 𝑧) βŠ† (π‘₯ ∩ 𝑦) β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
63, 5syl 17 . . . 4 (𝑧 βŠ† 𝑦 β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
76adantl 481 . . 3 ((𝐽 ∈ Top ∧ 𝑧 βŠ† 𝑦) β†’ ((π‘₯ ∩ 𝑦) = βˆ… β†’ (π‘₯ ∩ 𝑧) = βˆ…))
8 ineq2 4202 . . . . 5 (𝑦 = 𝑧 β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ 𝑧))
98eqeq1d 2729 . . . 4 (𝑦 = 𝑧 β†’ ((π‘₯ ∩ 𝑦) = βˆ… ↔ (π‘₯ ∩ 𝑧) = βˆ…))
109adantl 481 . . 3 ((𝐽 ∈ Top ∧ 𝑦 = 𝑧) β†’ ((π‘₯ ∩ 𝑦) = βˆ… ↔ (π‘₯ ∩ 𝑧) = βˆ…))
112, 7, 10opnneieqv 47842 . 2 (𝐽 ∈ Top β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
121, 11syl 17 1 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ((neiβ€˜π½)β€˜π·)(π‘₯ ∩ 𝑦) = βˆ… ↔ βˆƒπ‘¦ ∈ 𝐽 (𝐷 βŠ† 𝑦 ∧ (π‘₯ ∩ 𝑦) = βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3065   ∩ cin 3943   βŠ† wss 3944  βˆ…c0 4318  β€˜cfv 6542  Topctop 22769  neicnei 22975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22770  df-nei 22976
This theorem is referenced by:  sepnsepo  47855
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